constitutive viscoelastoplasticity

8/9/2019 Constitutive Viscoelastoplasticity

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VOI. 42 ~ u p p . SCIENCE IN CHINA (series D ) August 1999

Constitutive theories on viscoelastoplasticity anddamage of frozen soil *

HE Ping (IT ? , CHENG Guodong (@ El % ) an d ZHU Yaanlin (%%# )(State Key Laboratory of Frozen Soil Engineering, Lanzhou Institute of Glaciology and

Geocryology , Chinese Academy of Sciences, Lanzhou 730000, China)

Received February 8 , 1999

Abstract The constitutive theory on the viscoelastoplasticity and damage of frozen soil is based on the continuousmechanics and thermodynamics. The basic principles of the theory, dissipation potential function and damage model are

presented. The constitutive theory explains the mechanical properties of frozen soils under complicated stresses, espe-

cially under high confining pressures which make frozen soil harden and soften. The agreement between the calculated

results by the constitutive theory and the experimental results of triaxial creep of frozen soil is seen to be very good.

Keywords : viscoelastoplasticity , damage, unfrozen water, frozen soil creep.

The strength and creep properties of frozen soils have long been considered in engineering de-signs and construction~.With the increase in shaft depth by use of the frozen method, more and moreengineering problems related to frozen soil mechanics have been encountered. Especially, under high

ground stresses the frozen soil mechanical properties change grea tly. Experimental showthat the high confining pressures make frozen soils harden and also soften.On the one hand the high

confining pressure inc reases the friction between soil par ticles, while on the other hand it increasesunfrozen water content and decreases the cohesive force of ice with soil. In order to describe the com-plicated mechanical properties of frozen soil and to guide engineering designs and constructions, thereis a need to establish the constitutive theory of frozen soil.

1 Basicprinciple

The "damagen concept was first presented by Kachanov when he studied the fracture time of abar under tensile stress in 1958. S u b s e q u e n t l y, ~ e m a i t r e [ ~ - ~ ] ,rajcino vic[~- 'O ' , chabocher"] et

al. have developed the theory of damage mechanics. The theory was introduced to frozen soil mechan-ics in 1995[12' . Because of the complicated mechanical properties of frozen soil, the application ofdamage theory to frozen soil needs to be further studied.

The constitutive theory on viscoelastoplasticity and damage of frozen soil obeys the basic princi-p les of continuous mechan ics and t h e r m ~ d ~ n a m ic s [ ~ * '~ " ~ 'namely, the conservation of mass, balance

of momentum, conservation of energy and principle of entropy, i. .* Project supported by the National Natural Science Foundation of China (Grant No. 49571019) and Key Project of the Chinese

Academy of Sciences (Grant No. KZ952-J1-216) .


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S ~ P P . CONSTITUTIVE THEORIES ON VISCOELASTOPLASTICITY 39

where p is the mass density, vl is the component of the displacement velocity, akl s the component of

stress tensor, fi is the component of body force density, e is the internal energy density, Dk l s the

component of deformation velocity tensor, qk is the component of heat vector, j is the strength of heat.a( >

source, S is the entropy density, T is the thermodynamic temperatute, and ( ) , means -.axk

Here the free energy function (cp) s introduced:9 = e - Ts..

Substituting formula (6 ) into (4) and ( 5 ) , we have

p ( 6 + ~i + T S ) - oklDkl + qk,k - p~ = 0 ,

where 9 s the function of the elastic strain ( E : ~ ) , temperature ( T ) and internal variables ( a i ) .That is

9 = E L, T , Q ~ ) . ( 8 )The internal variable a i s defined as

( r , D , n W ) a i , (9 )where r , D and n w are the accumulative plastic strain with damage, the damage variable and un-

frozen water content, respectively. The plastic strain is defined from total strain by E % = . E M - & i l .The unfrozen water content ( n W ) s ice content ( n i ) dependent for saturated frozen soil in a closedsystem. That is iw i i 0 . Substituting (8) into (7) and considering that & Iland T are indepen-dent variables, we get

aakl = p 4 (elastic law),

s = - % (entropy law).a T (11)The following symbols are defined :

where R , Y , ,u and gk are the general forces corresponding to the state variables. We have

akFgl - R; - YLj - ,Aw gkqk a 0 . (13)Under isolation and no heat source conditions, (13) should be modified as follows:

akig l - & - a 3 0. (14)Formula (14) indicates that there exists a hypersurface in the general state space, and when the state

point changes from outside to inside the process does not dissipate energy. However, when from inside


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40 SCIENCE IN CHINA (Series D) Vol. 42

to outside, energy will be d issipa ted. The hypersurface called the dissipated potential function can beexpressed in general forces, that is

F (a kr , R , Y , p ) = 0 .The constitutive equations for the evolution of dissipated variables from the normality rule are written

I . a~dp, = A - 9a a,,

It is important to determine the actual dissipated potential function. The following form[lS1 ssuggested for frozen soil:

F = F p + F D + F,., (17 )

where F,, FD nd F; express the dissipated energies of the viscoplastic deformation, damage andthe process from ice to &zen water, respectively.

1 1 Plastic dissipated potential and evolution law

The plastic dissipated potential( F,) is suggested for frozen. soil as follows:

The equation expresses the influence of damage based on the hypotheses of effective stress andstrain equivalence[5*71- and the influence of unfrozen water content on the strengthening stress( R )and the initial yield stress ( 0 , ) . The definitions of the symbols in (18) are as follows:

- 1I I =

-- i J f f k k 9 (20 )

where Su , k l ,and D are the component of stress deviation tensor, the component of stress tensorand damage variable, respectively. Unfrozen water content depends on temperature and pressure:

n w = f ( T , I l ) ; ( 21thus

R ( n W , r ) = R ( T , l l , r ) , (22)

c y ( n W )= a ,( T, I l ) . (23)

Experimental results[161 how that the relationship between unfrozen water content and temperature isn W= a ~ - ' , (24)

where T is the absolute value of temperature. Considering the effect of the hydrostatic stress whichcau& the increase of unfrozen water content and frozen temperature,(24) can be replaced by

(25)


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SUPP CONSTITUTIVE THEORIE S ON VISCOELASTOPLASTICITY 4 1

where kw is a parameter which can be obtained by Clapeyron equation["]. Based on test results andconsidering ( 2 5 ) , ( 2 2 ) and ( 2 3 ) can be expressed by

k , l , q iR ( n w , r ) = k p ( T -

-)3 r n , ( 2 6 )

where kp q 1 , 9 2 , n and A are parameters corresponding to the soil type.From ( 1 6 ) - ( 1 8 ) , ( 2 2 ) and ( 2 3 ) we can obtain the laws of viscoplastic deformation develop-

ment:

r = A .

Here we define

where e$" is the component of plastic strain deviation tens or. Multiplying( 2 8 ) by itself and consider-ing ( 3 0 ), he following formula can be obtained:

The viscoplastic strain rate is related to the viscoplastic potential function, that isP = ~ J F ~ ) ~ . ( 3 2 )

1 2 Damage dissipation potential and damage evolution lawThe damage dissipation potential is suggested as follows:

The law of damage evolution is obtained by damage dissipated potential:

Based on experimental results and considering( 2 5 ), the function; g ( p n w ) s

where k , 3 and m are the parameters of soil type. In general, the damage of a material does not hap-pen until the deformation is up to a characteristic value called the damage initial strain( Pd) which isrelated to the state of viscoplastic potential, so the damage initial strain is considered as

P, = BF;, ( 3 6 )

where B and s are parameters.

1 .3 Elasticity lawElastic deformation law obeys Hooks' s law. That is


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42 SCIENCE IN CHINA (Series D ) Vol. 42

Formulae (28) , 29) , (31) , 32) , 34) and (37) consist of the viscoelastoplasticity and dam-

age constitutive equations of frozen soil.

2Example and conclusion

Figure 1 displays the calculated results by the viscoelastoplasticity and damage constitutive equa-

tions and the experimental results of the triaxial compressive creep of saturated frozen silt . It is foundby the comparison of the results that the constitutive theory is suitable to frozen soil. The viscoelasto-

plasticity and damage constitutive theory explains the effects of confining pressure which make frozen

soil harden by increasing the friction force between soil particles and soften by increasing unfrozen wa-

ter content and decreasing the cohesive force of ice with soil. The damage of internal structure of

frozen soil causes creep failure.

Fig. 1 . The experimental and calculated results of frozensilt under triaxial compressive creep. 1 , a 3

=1 5 M P a, u, - u 3 = 3 . 7 M Pa; 2 , u 3 = 1 8 M P a, a, -a, = 3 . 3 MPa; 3 , a 3 = 10 MPa,u, - u 3 = 2 . 8

MPa ; ---, experimental values, - calculated values.The calculated parameters can be obtained by uniaxial and triaxial compressive tests under con-

stant strain rates and at different temperatures. For saturated frozen silt with dry density of 1.6 x lo3

kg/m3 and water content of 25 % , the parameters are as follows:a = 0.003, n = 0.75, 6 = 10, q , = 0.31, q 2 = 0.9 , q3 = -0.396,

m = 0.239, A = 0.139, s = 0.77, k, = 0.075, k, = 0.174, k = 0 . 0 2 0 ,

References

1 Chamberlain, E . J . , Gmves, E . , erham, R . The mechanical behavior of frozen earth materials under high pressure triaxialtest conditions, &technique, 1972, 22(3) : 469.

2 Sayles, F. H . , Triaxial constant strain rate tests and triaxial creep tests on frozen Ottawa sand, inProceedings of 2 nd Inbema-


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SU PP . CONSTITUTIVE THEORIESON VISCOELASTOPLASTICITY 43

tional Permafrost Co$erence ( ed . Melnikov, P . . ) , Washington, D . C . : National Academy Press, 1973, 384-391 .Ma Wei , Wu Ziwang, Sheng Yu , The effects of confining pressure on th e strength of frozen soil, Chinese J o d f GeotechnicalEngineering (in Chinese) , 1995, 17(5) :7.Zhu Yuanlin, He Ping, Zhang Jiayi, Triaxial creep model of fmzen soil under dynamic loading, Progress in N a t w d Science,

1997, 7(4) : 465.Lemaitre, . , How to use damage mechanics, Nuclear Eng . and Design, 1984, 80 : 233.M t r e . A continuous damage mechanics model for ductile fracture, J . o Eng . M& & Technology, 985, 107 : 83.Lemaitre, J . , Damage Mechanics (in Chinese) , Beijing : Science Press, 1996, 46-186.h j c i n o v i c , D . , Fonseka, G . U , The continuous damage theory of brittle m aterials, Part I , 11, J . AppliedMechanics Tran s.of ASME, 1981, 48 : 809.Krajcinovic. D. , Statistical aspects of the continuous damage theory, Int . J . Solidr Structures , 1982, 18(7) : 551 .Krajcinovic, D . , Creep of stmctures- A continuous damage mechanics approach, J . of Structure Mechanics, 1983, 11 ( 1 ) :1.

Chaboche, J. L. , Continuum damage mechanics part I-General concepts, J . of Applied M e c h m h , 1988, 55 : 59.Miao Tiande, Wei Xuexia, Zhang Changqing, A study on creep of frozen soil by damage mechanics, Science in China (in Chi-

nese), Ser. B, 1995, 25(3) : 309.Kuang Zhenbang, Nonlinearity Contiuum Mechanics ( i n Chin ese ), Xi' an: Xi' an Jiaotong University Publishing House,1989,92-195.Wu Hongyao, Damage Mechanics (in Chinese) , Beijing : Defense Industry Publishing House, 1990, 1-34.Lou Zhiwen, Damage Mechanics (i n Chinese) , Xi' an: Xi' an Jiaotong University Publishing House,1991, 37-38.Xu Xuezu, Deng Yousheng, Experiment Study of Water Migration in FrozenSoil (i n Chin ese), Beijing: Science Press , 1991,23 .Wang Zhicheng, ThermodynamicsMd Statistical Physics (i n Chinese ), Beijing: Higher Education Press, 1980, 110-164.

constitutive viscoelastoplasticity

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